FEM Sensitivity Method for Uncertainty and Reconciliation Analyses

Parametric variation of large order finite element models is required for evaluation of uncertainty and test-analysis reconciliation analyses. Established procedures for computation of modal frequency and mode shape derivatives are widely used in such studies. While modal derivatives accurately describe sensitivities for small parametric changes, they may be ineffective when modal frequencies are closely spaced or repeated. Over the past decade, an alternative modal sensitivity procedure has been employed for modal test-analysis reconciliation, without rigorous proof of its validity. This procedure, based on definition of residual shape functions that augment baseline system mode shapes, produces reduced mass and stiffness (sensitivity) matrices. The resultant sensitivity formulation is extremely effective for computation of altered system modes associated with large parametric variations, regardless of the presence of closely-spaced or repeated modal frequencies. This paper provides a rigorous validity proof of the alternative modal sensitivity formulation. Residual shape functions are similar to well-known quasi-static residual vectors for both localized and highly distributed model changes. A simple illustrative example is provided to demonstrate effectiveness of the technique.